Question: Simplify and expand the following expression: $ \dfrac{4z - 10}{z - 5}+\dfrac{4z}{z + 5} $
Solution: In order to add expressions, they must have a common denominator. Get both fractions over a common denominator of $(z - 5)(z + 5)$ Multiply the first term by $\dfrac{z + 5}{z + 5}$ $ \begin{align*} \dfrac{4z - 10}{z - 5} \times \dfrac{z + 5}{z + 5} & = \dfrac{(4z - 10)(z + 5)}{(z - 5)(z + 5)} \\ & = \dfrac{4z^2 + 10z - 50}{(z - 5)(z + 5)}\end{align*} $ Multiply the second term by $\dfrac{z - 5}{z - 5}$ $ \begin{align*} \dfrac{4z}{z + 5} \times \dfrac{z - 5}{z - 5} & = \dfrac{(4z)(z - 5)}{(z + 5)(z - 5)} \\ & = \dfrac{4z^2 - 20z}{(z + 5)(z - 5)}\end{align*} $ Now we have: $ = \dfrac{4z^2 + 10z - 50}{(z - 5)(z + 5)} + \dfrac{4z^2 - 20z}{(z + 5)(z - 5)} $ Now both terms have a common denominator we can simply add the numerators: $ = \dfrac{4z^2 + 10z - 50 + 4z^2 - 20z}{(z - 5)(z + 5)} $ $ = \dfrac{8z^2 - 10z - 50}{(z - 5)(z + 5)}$ Expand the denominator: $ = \dfrac{8z^2 - 10z - 50}{z^2 - 25}$